Large Deviations in Ergodic Theory

Abstract

The classical example of a large deviation result is Cramer’s theorem. It tells us, in a contemporary formulation, that if Y₁, Y₂,… is a sequence of independent real valued random variables with identical distribution function F such that

$$ f(\theta ) = E[\exp \{ \theta Y_1 \} [ = \smallint \exp \{ \theta y\} F(dy)$$

is finite for all finite θ,and if Z_n = (Y₁) + Y₂ + … Y_n/n then

$$ \text{k}(\text{x})\, = \,\mathop {\sup }\limits_\theta \,[\theta x\, - \,\log \,f(\theta )] $$

satisfies

• (0.1) $\overline {_{n \to \infty }^{\lim } } \frac{1} {\text{n}}\,\log \,\text{P}[z_n \, \in \text{A}]\, \leqslant \text{ } - \inf \text{ }k(a):a \in \text{A}$ A closed

and

• (0.2) $\overline {_{n \to \infty }^{\lim } } \frac{1}{\text{n}}\,\log \,\text{P}[z_n \, \in \text{A}]\, \geqslant \text{ } - \inf \{ k(a):a \in \text{A}\}$ A Open.